cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A371363 Expansion of (1/x) * Series_Reversion( x * (1-3*x)^3 / (1-2*x) ).

Original entry on oeis.org

1, 7, 85, 1261, 20788, 365845, 6731758, 127938625, 2491921516, 49480794460, 997897366717, 20384025765619, 420869454302620, 8769197604091246, 184151509243984300, 3893585866824069577, 82817275938125471548, 1770880435886367151060
Offset: 0

Views

Author

Seiichi Manyama, Mar 19 2024

Keywords

Crossrefs

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serreverse(x*(1-3*x)^3/(1-2*x))/x)
    
  • PARI
    a(n) = sum(k=0, n, 2^(n-k)*binomial(3*n+k+2, k)*binomial(3*n+1, n-k))/(n+1);

Formula

a(n) = (1/(n+1)) * Sum_{k=0..n} 2^(n-k) * binomial(3*n+k+2,k) * binomial(3*n+1,n-k).

A371365 Expansion of (1/x) * Series_Reversion( x * (1-4*x)^3 / (1-3*x) ).

Original entry on oeis.org

1, 9, 141, 2701, 57513, 1307553, 31083925, 763267077, 19208408721, 492817411705, 12842067417309, 338956669920189, 9042967461581753, 243464712274093713, 6606427290991922277, 180492205687604057013, 4960765361688213527073
Offset: 0

Views

Author

Seiichi Manyama, Mar 19 2024

Keywords

Crossrefs

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serreverse(x*(1-4*x)^3/(1-3*x))/x)
    
  • PARI
    a(n) = sum(k=0, n, 3^(n-k)*binomial(3*n+k+2, k)*binomial(3*n+1, n-k))/(n+1);

Formula

a(n) = (1/(n+1)) * Sum_{k=0..n} 3^(n-k) * binomial(3*n+k+2,k) * binomial(3*n+1,n-k).

A369024 Expansion of (1/x) * Series_Reversion( x * (1-2*x)^4 / (1-x) ).

Original entry on oeis.org

1, 7, 81, 1135, 17617, 291479, 5038177, 89901023, 1643514849, 30623478951, 579444828465, 11103818394447, 215053322179121, 4202849976054583, 82778942956393409, 1641477474636943295, 32743892109730116801, 656612555241354578759, 13228883898856161274129
Offset: 0

Views

Author

Seiichi Manyama, Jan 12 2024

Keywords

Crossrefs

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serreverse(x*(1-2*x)^4/(1-x))/x)
    
  • PARI
    a(n, s=1, t=4, u=-1) = sum(k=0, n\s, binomial(t*(n+1)+k-1, k)*binomial((t+u+1)*(n+1)-(s-1)*k-2, n-s*k))/(n+1);

Formula

a(n) = (1/(n+1)) * Sum_{k=0..n} binomial(4*n+k+3,k) * binomial(4*n+2,n-k).
D-finite with recurrence -2*(462919*n-251445)*(4*n+1) *(2*n+1)*(4*n+3) *(n+1)*a(n) +(625365036*n^5 +403579400*n^4 -437229300*n^3 +49132810*n^2 -20878971*n +3771675)*a(n-1) +(484851248*n^5 -3077382030*n^4 +7964893000*n^3 -10232074140*n^2 +6398384592*n -1533654945)*a(n-2) +(652184*n-451475)*(4*n-9) *(n-2)*(4*n-7)*(2*n-3)*a(n-3)=0. - R. J. Mathar, Jan 25 2024
Showing 1-3 of 3 results.